Metamath Proof Explorer


Theorem supicclub2

Description: The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019)

Ref Expression
Hypotheses supicc.1 ( 𝜑𝐵 ∈ ℝ )
supicc.2 ( 𝜑𝐶 ∈ ℝ )
supicc.3 ( 𝜑𝐴 ⊆ ( 𝐵 [,] 𝐶 ) )
supicc.4 ( 𝜑𝐴 ≠ ∅ )
supiccub.1 ( 𝜑𝐷𝐴 )
supicclub2.1 ( ( 𝜑𝑧𝐴 ) → 𝑧𝐷 )
Assertion supicclub2 ( 𝜑 → sup ( 𝐴 , ℝ , < ) ≤ 𝐷 )

Proof

Step Hyp Ref Expression
1 supicc.1 ( 𝜑𝐵 ∈ ℝ )
2 supicc.2 ( 𝜑𝐶 ∈ ℝ )
3 supicc.3 ( 𝜑𝐴 ⊆ ( 𝐵 [,] 𝐶 ) )
4 supicc.4 ( 𝜑𝐴 ≠ ∅ )
5 supiccub.1 ( 𝜑𝐷𝐴 )
6 supicclub2.1 ( ( 𝜑𝑧𝐴 ) → 𝑧𝐷 )
7 iccssxr ( 𝐵 [,] 𝐶 ) ⊆ ℝ*
8 1 2 3 4 supicc ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ( 𝐵 [,] 𝐶 ) )
9 7 8 sselid ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ℝ* )
10 3 7 sstrdi ( 𝜑𝐴 ⊆ ℝ* )
11 10 5 sseldd ( 𝜑𝐷 ∈ ℝ* )
12 10 sselda ( ( 𝜑𝑧𝐴 ) → 𝑧 ∈ ℝ* )
13 11 adantr ( ( 𝜑𝑧𝐴 ) → 𝐷 ∈ ℝ* )
14 12 13 xrlenltd ( ( 𝜑𝑧𝐴 ) → ( 𝑧𝐷 ↔ ¬ 𝐷 < 𝑧 ) )
15 6 14 mpbid ( ( 𝜑𝑧𝐴 ) → ¬ 𝐷 < 𝑧 )
16 15 nrexdv ( 𝜑 → ¬ ∃ 𝑧𝐴 𝐷 < 𝑧 )
17 1 2 3 4 5 supicclub ( 𝜑 → ( 𝐷 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧𝐴 𝐷 < 𝑧 ) )
18 16 17 mtbird ( 𝜑 → ¬ 𝐷 < sup ( 𝐴 , ℝ , < ) )
19 9 11 18 xrnltled ( 𝜑 → sup ( 𝐴 , ℝ , < ) ≤ 𝐷 )